{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 简单理解 LT-DF SOS RMP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 创建时间：2019-10-17；最后修改：2021-06-12"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这一节承接上一节对 DF-MP2 的讨论。我们知道 DF-MP2 的计算耗时尽管比传统 MP2 低但仍然是 $O(N^5)$；这一节作的 LT-DF SOS 近似则可以确实地将计算复杂度降低为 $O(N^4)$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":::{note}\n",
    "\n",
    "LT-DF SOS MP2 的写法并不是目前通用的写法；可能目前也没有通用的写法。\n",
    "\n",
    "该名称的含义是 Laplace-Transformation Density-Fitting Scaled Opposite-Spin Second Order Moller-Plesset。\n",
    "\n",
    "不要被这么长的名称吓到。除了 LT 之外，其它的过程或者非常容易实现 (SOS)，或者是我们已经理解的内容 (DF, MP2)。\n",
    "\n",
    ":::"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "\n",
    "import numpy as np\n",
    "import scipy\n",
    "import quadpy\n",
    "from functools import partial\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from pyscf import gto, scf, mp, df\n",
    "\n",
    "np.einsum = partial(np.einsum, optimize=[\"greedy\", 1024 ** 3 * 2 / 8])\n",
    "np.set_printoptions(8, linewidth=150, suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在这篇文档中，我们将会将测评体系更换为乙烯分子。这主要是为了与 Jung et al. [^Jung-Head-Gordon.JCP.2004.121] 文章中 Table I 的数据作比较。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":::{admonition} 冻核近似说明\n",
    "\n",
    "这篇文章采用了 MP2 的冻核近似 (Frozen Core)；因此我们之后的 MP2 计算也需要相应地采用冻核近似。据我所知 PySCF 没有明确地写出冻核近似代码。因此，我们需要手动指定冻核。对于乙烯分子，两个碳原子有需要被冻结的 1s 轨道，因此被冻结的占据轨道数为 2；而非占轨道的冻结数为零。\n",
    "\n",
    "程序中使用 `sf` 表示冻结后的占据轨道分割，`Cf` 与 `ef` 表示冻结后的分子轨道系数与轨道能量。为了符号简便起见，其它的程序符号基本沿用上一篇文档。\n",
    "\n",
    ":::"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":::{admonition} 分子结构说明\n",
    "\n",
    "目前使用的分子结构是在 MP2/6-31G* 下优化的乙烯分子。优化使用 Gaussian 09 rev B01 实现。尽管能重复 Jung et al. [^Jung-Head-Gordon.JCP.2004.121] 文章中乙烯分子的结果，但会在相关能的第 4 位上有些微差距。我认为这并不影响讨论。\n",
    "\n",
    "MP2 结构优化输入卡 {download}`assets/C2H4-opt.gjf` QCISD(T) 能量输入卡 {download}`assets/C2H4-eng.gjf`\n",
    "\n",
    ":::"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "mol = gto.Mole()\n",
    "mol.atom = \"\"\"\n",
    "C   -0.668188207     0.000000075    -0.0000028526\n",
    "C    0.668188207     0.0000000727    0.0000028526\n",
    "H   -1.2382848349   -0.0000000262   -0.9232789581\n",
    "H   -1.2382926466   -0.0000000483    0.9232684342\n",
    "H    1.2382848349   -0.0000000255    0.9232789581\n",
    "H    1.2382926466   -0.0000000477   -0.9232684342\n",
    "\"\"\"\n",
    "mol.basis = \"cc-pVTZ\"\n",
    "mol.verbose = 0\n",
    "mol.build()\n",
    "\n",
    "natm = mol.natm\n",
    "nao = nmo = mol.nao\n",
    "nocc = mol.nelec[0]\n",
    "nvir = nmo - nocc\n",
    "so, sv, sa = slice(0, nocc), slice(nocc, nmo), slice(0, nmo)\n",
    "sf = slice(2, nocc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考值"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 参考值与 SOS-MP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据 Jung et al. [^Jung-Head-Gordon.JCP.2004.121] 文章的结果，我们给出下述参考值字典 `ref`；每个键值对应的数值单位是 mH (milli-Hartree)。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "ref = {\n",
    "    \"QCISDT\": -375.4,\n",
    "    \"SS\": -35.5,\n",
    "    \"OS\": -264.7,\n",
    "    \"MP2\": -375.4 * 0.894,\n",
    "    \"SCS\": -375.4 * 0.909,\n",
    "    \"SOS\": -375.4 * 0.917,\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "其中，键值 `MP2` (Second Order Moller-Plesset), `SCS` (Spin-Component Scaled) 与 `SOS` (Scaled Opposite-Spin) 原则上都可以通过 `SS` (Same-Spin) 与 `OS` (Opposite-Spin) 对相关能的贡献给出："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "E_\\mathrm{MP2, corr} = 2 E_\\mathrm{SS} + E_\\mathrm{OS}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-335.6076\n",
      "-335.7\n"
     ]
    }
   ],
   "source": [
    "print(ref[\"MP2\"])\n",
    "print(ref[\"SS\"] * 2 + ref[\"OS\"] * 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "E_\\text{SCS-MP2, corr} = 2 \\times \\frac{1}{3} E_\\mathrm{SS} + \\frac{6}{5} E_\\mathrm{OS}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-341.23859999999996\n",
      "-341.3066666666667\n"
     ]
    }
   ],
   "source": [
    "print(ref[\"SCS\"])\n",
    "print(ref[\"SS\"] * 2 * 1/3 + ref[\"OS\"] * 6/5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "E_\\text{SOS-MP2, corr} = 1.3 E_\\mathrm{OS}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-344.2418\n",
      "-344.11\n"
     ]
    }
   ],
   "source": [
    "print(ref[\"SOS\"])\n",
    "print(ref[\"OS\"] * 1.3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": true
   },
   "source": [
    "关于这些方法的细节，我们不在这里细究。在这篇文档中，我们只关心到 SOS 方法只使用了 MP2 相关能中 OS 部分的贡献。这对我们后面的讨论至关重要。我们在这篇文档将要重复的就是 SOS-MP2。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SCF 计算"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在继续文档以前，我们需要给出乙烯分子的各种计算结果。大多数代码记号与公式都可以与前两篇文档比对。补充出来的记号除了冻结轨道近似的部分之外，还有\n",
    "\n",
    "* $D_i^a$ `D_ia`：$\\varepsilon_i - \\varepsilon_a$\n",
    "\n",
    "由于我们关心的是 MP2 部分的 DF，因此自洽场即使不使用 DF 也不影响我们的讨论。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "scf_normal = scf.RHF(mol).run()\n",
    "mp2_normal = mp.MP2(scf_normal).run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "C, e = scf_normal.mo_coeff, scf_normal.mo_energy\n",
    "Cf, Cv = C[:, sf], C[:, sv]\n",
    "ef, ev = e[sf], e[sv]\n",
    "D_iajb = ef[:, None, None, None] - ev[None, :, None, None] + ef[None, None, :, None] - ev[None, None, None, :]\n",
    "D_ia = ef[:, None] - ev[None, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SS/OS 计算"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "MP2 能量除了使用上一份文档的 $E_\\mathrm{MP2, corr} = T_{ij}^{ab} t_{ij}^{ab} D_{ij}^{ab}$ 之外，还可以使用下述的公式计算：\n",
    "\n",
    "$$\n",
    "E_\\mathrm{MP2, corr} = \\frac{1}{D_{ij}^{ab}} (ia|jb) \\big[ 2 (ia|jb) - (ib|ja) \\big]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上式可以拆分为\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "E_\\mathrm{SS} &= \\frac{1}{2} \\frac{1}{D_{ij}^{ab}} (ia|jb) \\big[ (ia|jb) - (ib|ja) \\big] \\\\\n",
    "E_\\mathrm{OS} &= \\frac{1}{D_{ij}^{ab}} (ia|jb)^2 \\\\\n",
    "E_\\mathrm{MP2, corr} &= 2 E_\\mathrm{SS} + E_\\mathrm{OS}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "其拆分依据是通过自旋轨道下的 MP2 表达式产生而来，这里不作展开。为了计算 $E_\\mathrm{OS}$，除了 $D_{ij}^{ab}$ `(D_iajb)` 之外，我们还需要 $(ia|jb)$。该变量储存于 `eri_iajb` 中："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "eri_iajb = np.einsum(\"ui, va, uvkl, kj, lb -> iajb\", Cf, Cv, mol.intor(\"int2e\"), Cf, Cv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "随后我们就可以计算 $E_\\mathrm{SS}$ `E_SS_normal` 与 $E_\\mathrm{OS}$ `E_OS_normal` 了："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E_SS       in mH:  -35.46668203417094\n",
      "E_OS       in mH:  -264.8091129392841\n",
      "E_MP2_corr in mH:  -335.742477007626\n"
     ]
    }
   ],
   "source": [
    "E_SS_normal = 0.5 * (eri_iajb * (eri_iajb - eri_iajb.swapaxes(-1, -3)) / D_iajb).sum()\n",
    "E_OS_normal = (eri_iajb ** 2 / D_iajb).sum()\n",
    "print(\"E_SS       in mH: \", E_SS_normal * 1000)\n",
    "print(\"E_OS       in mH: \", E_OS_normal * 1000)\n",
    "print(\"E_MP2_corr in mH: \", (E_OS_normal + 2 * E_SS_normal) * 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这篇文档的主要目标并非重复 $E_\\mathrm{MP2, corr}$，而是重复 $E_\\text{SOS-MP2, corr}$ `E_SOS_normal`："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E_SOS_corr in mH:  -344.25184682106936\n"
     ]
    }
   ],
   "source": [
    "E_SOS_normal = 1.3 * E_OS_normal\n",
    "print(\"E_SOS_corr in mH: \", E_SOS_normal * 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Laplace 变换 (LT) 原理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $x^{-1}$ LT 数值积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在这里，我们不讨论更为广泛的 LT 的原理，只讨论与 MP2 计算相关的 $x^{-1} = (D_{ij}^{ab})^{-1}$ 的计算问题。对于 LT 中涉及到 MP2 的问题会放在下一小节中讨论；这里我们只需要知道乙烯分子 $x$ 的取值范围大约会是"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "|D_iajb| max in Hartree:  31.40203636576501\n",
      "|D_iajb| min in Hartree:  1.0534258481856735\n"
     ]
    }
   ],
   "source": [
    "print(\"|D_iajb| max in Hartree: \", np.abs(D_iajb).max())\n",
    "print(\"|D_iajb| min in Hartree: \", np.abs(D_iajb).min())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x^{-1}$ 可以通过下述方式获得：\n",
    "\n",
    "$$\n",
    "x^{-1} = \\int_0^{+ \\infty} e^{- x t} \\, \\mathrm{d} t\n",
    "$$\n",
    "\n",
    "根据格点积分的原理，我们可以将上式在格点 $\\{g\\}$ 下重新写为\n",
    "\n",
    "$$\n",
    "x^{-1} \\simeq w_g e^{- x t_g}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在我们从程序的角度实现上述格点积分。我们知道，$e^{-xt}$ 函数在零点处的值较大，但在远离零点处的值较小。同时，由于 $e^{-xt}$ 本身就是一种指数形式，因此我们容易想到使用指数的方式构造格点。令坐标变量 $t_g$ `grid_points`\n",
    "\n",
    "$$\n",
    "t_g = a^g\n",
    "$$\n",
    "\n",
    "那么对应的格点权重 $w_g$ `grid_weights` 应当接近\n",
    "\n",
    "$$\n",
    "w_g \\simeq \\frac{\\partial}{\\partial g} t_g = \\log a \\cdot a^g\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在这里，我们取 $a = 2.5$，而 $g \\in \\left[ -10, 5 \\right)$。这种取法一定程度上是任意的，但考虑到我们希望对 $x \\in (0.1, 100)$ 的区间，特别是靠近 $x \\in (0.1, 2)$ 的区域有比较良好的近似 (MP2 中主要产生贡献的部分在分母 $x$ 值偏小的区域)，因此这些参数并不是彻底任意地选取的。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00001678,  0.00004194,  0.00010486,  0.00026214,  0.00065536,  0.0016384 ,  0.004096  ,  0.01024   ,  0.0256    ,  0.064     ,  0.16      ,\n",
       "        0.4       ,  1.        ,  2.5       ,  6.25      , 15.625     , 39.0625    , 97.65625   ])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grid_points = 2.5 ** np.arange(-12, 6)\n",
    "grid_weights = np.log(2.5) * grid_points\n",
    "grid_points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "举例来说，如果我们要用格点近似 $2^{-1} = 0.5 \\simeq w_g e^{- 2 t_g}$，那么程序可以编写如下："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0001747291147214"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(grid_weights * np.exp(- 1 * grid_points)).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这应该是一个比较有效的近似了。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### LT 数值积分精度"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于更广泛的数值精度分析，我们在上述 LT 格点的条件下，给出如下的图像："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        fig.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width, fig.canvas.height);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"\" width=\"640\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Ratio of $\\\\sum_g w_g e^{-x t_g}$ to $1/x$')"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "x_axis = np.logspace(-2, 3, 100)\n",
    "y_axis = (grid_weights * np.exp(- x_axis[:, None] * grid_points)).sum(axis=-1) * x_axis\n",
    "ax.plot(x_axis, y_axis)\n",
    "ax.plot([1e-2, 1e3], [1.001, 1.001], color=\"C1\", linestyle=\":\")\n",
    "ax.plot([1e-2, 1e3], [1, 1], color=\"black\", linestyle=\":\")\n",
    "ax.plot([1e-2, 1e3], [0.999, 0.999], color=\"C1\", linestyle=\":\")\n",
    "ax.plot([1e-1, 1e-1], [0, 2], color=\"C2\", linestyle=\":\")\n",
    "ax.plot([1e2, 1e2], [0, 2], color=\"C2\", linestyle=\":\")\n",
    "ax.set_ylim(0.99,1.002)\n",
    "ax.set_xscale(\"log\")\n",
    "\n",
    "ax.set_xlabel(\"$x$\")\n",
    "ax.set_ylabel(\"Relative error\")\n",
    "ax.set_title(\"Ratio of $\\sum_g w_g e^{-x t_g}$ to $1/x$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上图表示的是在不同 $x$ 取值下，我们所给出的格点积分 $w_g e^{-x t_g}$ 的精度；在 $x \\in (0.1, 100)$ 的区间内我们仍然能保证 $0.1\\%$ 的精度，但超过这些区域之后，结果就会迅速地变差。\n",
    "\n",
    "我们知道，若分子呈现近简并的情形，那么 $x = D_{ij}^{ab}$ 会非常小；在这种情况下，LT 所提供的格点积分就很可能无法正确地描述分母 $D_{ij}^{ab}$ 的行为，使得本来在近简并就无法正常计算的 MP2 的情况雪上加霜。但若分子的 HOMO/LUMO 的能级差足够大并且大于 $0.1$ 个 Hartree，那么我们应当预期当前格点下的 LT 近似可以几乎精确地给出相关能。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### LT SOS MP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在我们就可以用上刚才的技术，对 LT SOS MP2 进行计算。这并不会提升任何计算效率；这里只是用来验证我们方才作的 LT 近似的有效性。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在 LT 近似下，SOS 能量可以表达为\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "E_\\text{SOS-MP2, corr} &= 1.3 \\frac{1}{D_{ij}^{ab}} (ia|jb)^2 \\\\\n",
    "&= - 1.3 (ia|jb)^2 \\int_0^{+\\infty} e^{D_{ij}^{ab} t} \\, \\mathrm{d} t \\\\\n",
    "&\\simeq -1.3 w_g (ia|jb)^2e^{D_{ij}^{ab} t_g}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这里的公式的正负号可能不太直观。这是因为在这篇文档的记号下，$D_{ij}^{ab} < 0$。我们通过如下代码，给出 LT SOS MP2 能量："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LT-SOS-MP2 corr      in mH:  -344.2429384155852\n",
      "Deviation to SOS-MP2 in mH:  0.00890840548417593\n"
     ]
    }
   ],
   "source": [
    "E_SOS_LT = - 1.3 * np.einsum(\"g, iajb, giajb ->\", grid_weights, eri_iajb**2, np.exp(D_iajb * grid_points[:, None, None, None, None]))\n",
    "print(\"LT-SOS-MP2 corr      in mH: \", E_SOS_LT * 1000)\n",
    "print(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LT - E_SOS_normal) * 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以发现这是一个有效的近似。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LT-DF SOS MP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DF 环境设置"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在继续文档之前，我们需要对 DF 的环境作一些简单的设置。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "auxmol = mol.copy()\n",
    "auxmol.basis = \"cc-pvtz-ri\"\n",
    "auxmol.build()\n",
    "\n",
    "nao_df = auxmol.nao"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "int2c2e = auxmol.intor(\"int2c2e\")\n",
    "int2c2e.shape\n",
    "int3c2e = df.incore.aux_e2(mol, auxmol)\n",
    "\n",
    "int2c2e_half = scipy.linalg.cholesky(int2c2e, lower=True)\n",
    "V_df_mp2 = scipy.linalg.solve_triangular(int2c2e_half, int3c2e.reshape(-1, nao_df).T, lower=True)\\\n",
    "               .reshape(nao_df, nao, nao).transpose((1, 2, 0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "尽管这里的 $V_{ia, P}$ `V_df_ia` 与前一篇文档中的 $V_{ia, P}^\\mathrm{MP2}$ 是相同的，但需要注意到这里的轨道 $i$ 是价层的占据轨道，而不能是冻结的占据轨道。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "V_df_ia = np.einsum(\"uvP, ui, va -> iaP\", V_df_mp2, Cf, Cv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### LT-DF SOS MP2 表达式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们对上面提到的 LT SOS MP2 的近似表达式再作 DF 近似：\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "E_\\text{SOS-MP2, corr} &\\simeq -1.3 w_g (ia|jb)^2 e^{D_{ij}^{ab} t_g} \\\\\n",
    "&\\simeq -1.3 w_g V_{ia, P} V_{jb, P} V_{ia, Q} V_{jb, Q} e^{D_{ij}^{ab} t_g} \\\\\n",
    "&= -1.3 w_g V_{ia, P} V_{jb, P} V_{ia, Q} V_{jb, Q} e^{D_i^a t_g} e^{D_j^b t_g}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "注意到最后一个等号利用到 $D_{ij}^{ab} = D_i^a + D_j^b$。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LT-DF-SOS-MP2 corr   in mH:  -344.1316103156453\n",
      "Deviation to SOS-MP2 in mH:  0.12023650542408726\n"
     ]
    }
   ],
   "source": [
    "E_SOS_LTDF = - 1.3 * np.einsum(\"g, iaP, jbP, iaQ, jbQ, gia, gjb ->\",\n",
    "                               grid_weights, V_df_ia, V_df_ia, V_df_ia, V_df_ia,\n",
    "                               np.exp(D_ia * grid_points[:, None, None]),\n",
    "                               np.exp(D_ia * grid_points[:, None, None]))\n",
    "print(\"LT-DF-SOS-MP2 corr   in mH: \", E_SOS_LTDF * 1000)\n",
    "print(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LTDF - E_SOS_normal) * 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们认为这基本重复出了原始的 SOS MP2 能量了。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 简化公式表达"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "显然上面的公式表达太过于繁杂。在实际实现时，一般先将其中的三个张量缩并为 `X_gPQ`：\n",
    "\n",
    "$$\n",
    "X_{gPQ} = V_{ia, P} V_{ia, Q} e^{D_i^a t_g}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_gPQ = np.einsum(\"iaP, iaQ, gia -> gPQ\", V_df_ia, V_df_ia, np.exp(D_ia * grid_points[:, None, None]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "那么 LT-DF 近似后的 SOS MP2 表达式可以简写为\n",
    "\n",
    "$$\n",
    "E_\\text{SOS-MP2, corr} \\simeq -1.3 w_g X_{gPQ}^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LT-DF-SOS-MP2 corr   in mH:  -344.13161031563544\n",
      "Deviation to SOS-MP2 in mH:  0.12023650543391273\n"
     ]
    }
   ],
   "source": [
    "E_SOS_LTDF_simp = - 1.3 * np.einsum(\"g, gPQ ->\", grid_weights, X_gPQ**2)\n",
    "print(\"LT-DF-SOS-MP2 corr   in mH: \", E_SOS_LTDF_simp * 1000)\n",
    "print(\"Deviation to SOS-MP2 in mH: \", (E_SOS_LTDF_simp - E_SOS_normal) * 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 计算效率比较"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这里我们对 SOS MP2，LT SOS MP2 与 LT-DF SOS MP2 的计算效率进行比较，工具是 `np.einsum_path`。我们假定内存的空间是足够多的，并且不考虑为生成各种积分、或各种 DF 三中心积分所需要耗费的时间。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**普通 MP2**\n",
    "\n",
    "效率耗时最大的部分在双电子积分转换过程。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  ui,va,uvkl,kj,lb->iajb\n",
      "         Naive scaling:  8\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  3.801e+14\n",
      "  Optimized FLOP count:  2.487e+09\n",
      "   Theoretical speedup:  152841.286\n",
      "  Largest intermediate:  9.365e+06 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   5               uvkl,ui->iklv                      va,kj,lb,iklv->iajb\n",
      "   5               iklv,kj->ijlv                         va,lb,ijlv->iajb\n",
      "   5               ijlv,va->ijal                            lb,ijal->iajb\n",
      "   5               ijal,lb->iajb                               iajb->iajb\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"ui, va, uvkl, kj, lb -> iajb\",\n",
    "    Cf, Cv, mol.intor(\"int2e\"), Cf, Cv,\n",
    "    optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1])  # Given 2GB memory space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**DF SOS MP2**\n",
    "\n",
    "效率耗时最大的部分在从 3c2e 积分生成双电子积分过程。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  iaP,jbP->iajb\n",
      "         Naive scaling:  5\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  2.368e+08\n",
      "  Optimized FLOP count:  2.368e+08\n",
      "   Theoretical speedup:  1.000\n",
      "  Largest intermediate:  4.199e+05 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   5               jbP,iaP->iajb                               iajb->iajb\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"iaP, jbP -> iajb\",\n",
    "    V_df_ia, V_df_ia, \n",
    "    optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1])  # Given 2GB memory space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**LT-DF SOS MP2**\n",
    "\n",
    "效率耗时最大的部分在生成中间张量 $X_{gPQ}$ 的过程。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  iaP,iaQ,gia->gPQ\n",
      "         Naive scaling:  5\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  2.783e+09\n",
      "  Optimized FLOP count:  1.858e+09\n",
      "   Theoretical speedup:  1.497\n",
      "  Largest intermediate:  3.289e+06 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   4               gia,iaP->igaP                            iaQ,igaP->gPQ\n",
      "   5               igaP,iaQ->gPQ                                 gPQ->gPQ\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"iaP, iaQ, gia -> gPQ\",\n",
    "    V_df_ia, V_df_ia, np.exp(D_ia * grid_points[:, None, None]),\n",
    "    optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1])  # Given 2GB memory space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于这个分子而言，放开优化张量缩并过程，结果反而会回到类似于 DF MP2 的缩并过程："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  g,iaP,jbP,iaQ,jbQ,gia,gjb->\n",
      "         Naive scaling:  7\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  4.207e+12\n",
      "  Optimized FLOP count:  4.892e+08\n",
      "   Theoretical speedup:  8600.264\n",
      "  Largest intermediate:  4.199e+05 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   3                  gia,g->iga                iaP,jbP,iaQ,jbQ,gjb,iga->\n",
      "   5               jbP,iaP->ijab                   iaQ,jbQ,gjb,iga,ijab->\n",
      "   5               jbQ,iaQ->ijab                      gjb,iga,ijab,ijab->\n",
      "   4             ijab,ijab->ijab                           gjb,iga,ijab->\n",
      "   5               ijab,gjb->iga                                iga,iga->\n",
      "   3                   iga,iga->                                       ->\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"g, iaP, jbP, iaQ, jbQ, gia, gjb ->\",\n",
    "    grid_weights, V_df_ia, V_df_ia, V_df_ia, V_df_ia,\n",
    "    np.exp(D_ia * grid_points[:, None, None]),\n",
    "    np.exp(D_ia * grid_points[:, None, None]),\n",
    "    optimize=[\"greedy\", 1024 ** 3 * 2 / 8])[1])  # Given 2GB memory space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**简单总结**\n",
    "\n",
    "事实上，LT-DF SOS MP2 的优势并不体现在小分子的计算上，这从刚才的效率分析上就能看出来。但是，对于大分子 (占据电子数较多的情况，而非基组非常庞大的情况)，LT-DF MP2 就有不小的优势。这是因为耗时最大的部分的时间复杂度分别是\n",
    "\n",
    "* LT-DF SOS MP2: $O(n_\\mathrm{grid} n_\\mathrm{nocc} n_\\mathrm{nvir} n_\\mathrm{aux}^2)$\n",
    "\n",
    "* DF SOS MP2: $O(n_\\mathrm{occ}^2 n_\\mathrm{vir}^2 n_\\mathrm{aux})$\n",
    "\n",
    "我们可能会认为 $n_\\mathrm{occ} < n_\\mathrm{vir}$，而 $n_\\mathrm{aux}$ 在基组较大时是 $n_\\mathrm{vir}$ 的 3 倍左右。由于 $n_\\mathrm{grid}$ 不随体系增大而变化，因此在 $n_\\mathrm{grid} \\ll n_\\mathrm{occ}$ 的情况下，LT 的优势会凸显。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 备注"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这份文档颜文杰提供了 QChem 代码中关于 7 点 LT 格点的支持；尽管最终文档使用了自定义的等比级数格点，但 7 点 LT 格点帮助了文档的完成的过程。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[^Jung-Head-Gordon.JCP.2004.121]: Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled Opposite-Spin Second Order Møller–Plesset Correlation Energy: An Economical Electronic Structure Method. *J. Chem. Phys.* **2004**, *121* (20), 9793–9802. doi: [10.1063/1.1809602](https://doi.org/10.1063/1.1809602)."
   ]
  }
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